Concentrated Actions on Cuspidate Plane Bodies

Examples are provided of locally integrable vector fields in the plane such that, when restricted to a cuspidate set, require an action concentrated at the cusp to be balanced, no matter what the order of the cusp. Examples are also given of vector fields of the same sort that, in spite of their cusp singularity, are balanced by everywhere null Neumann data at the boundary. All these examples are given explicit thermomechanical interpretations. Collectively, they are meant to support the view that a more general continuum-mechanical theory of contact interactions is needed to complement the standard theory, namely, a theory allowing both for point- or line-concentrated body/environment interactions and for internal interactions concentrated on body parts of zero volume measure.